2The number nf of active flavours depends on the scale µ we are considering. Only the quarks with masses smaller than the scale µ affect the beta function.
and it can be rewritten in a way so that it depends on only one parameter g2(µ) = (4π)2 (113 Nc−23nf)log( µ 2 Λ2 QCD) . (2.5)
At large energies µ the coupling tends to zero, which means that the quarks tend to behave as free particles in this regime. This phenomenon is called asymptotic freedom [1, 2] and is characteristic of all non abelian gauge theories with nf < 112Nc,
at least in four dimensions. On the other hand, for small values of µ the coupling becomes very large and results in the colour confinement of quarks, where quarks form bound states. ΛQCD is the scale at which the coupling diverges to infinity and
its value is estimated to ΛQCD ∼ 200 MeV. Note that the size of the hadrons is of
the order Λ−1QCD [15, 16].
Because QCD becomes asymptotically free at large energies, perturbation theory is a valid method for performing calculations in this regime. At small energies though the coupling becomes very large and perturbation theory can not be applied. Different tools are required for exploring the physics of the strong coupling. One such tool is introduced and applied in this thesis and it is called gauge/gravity duality. It is used to understand various strongly coupled field theories.
2.3
Wilsonian Approach
Before proceeding any further with QCD, let us introduce a different approach to renormalisation theory. As already mentioned, divergences appear in quantum field theories in some quantities like the masses and couplings. These divergence are a result of high energy momentum modes in the loop corrections. Apart from the infinities in those specific quantities, the high momentum modes do not really affect other computations in the theory and that is because generally the fields at
different energy scales are independent degrees of freedom. A very geometrical and intuitive picture is the one given by Wilson [5], who instead of using the usual renormalisation techniques for removing the divergences in the theory, he described the physics at different energies through scale dependent quantities.
In the path integral formulation of quantum field theory, the degrees of freedom of the theory are variables of integration. In the following expression, the integration variables are the Fourier components of the fields φ(k)
Z[J] = Z DφeRi[L+Jφ]= Y k≤Λ Z dφ(k)eRi[L+Jφ], (2.6) where L is the bare Lagrangian, J is the source and Λ is the momentum scale at which the bare Lagrangian is define at. If we focus on phenomena which are related to some specific momentum scale Λ0, where Λ0 ≤ Λ, then, taking into account the
fact that phenomena in different energy scales are decoupled, a new Lagrangian can be define which is valid for k ≤ Λ0. The new Lagrangian will result from the bare Lagrangian plus some correction which arise when all the modes with momentum
Λ0 ≤ k ≤ Λ, in the path integral, are integrated out. This Lagrangian is called the
effective Lagrangian and is defined (Euclidean signature) as
Z[J] = Y k≤Λ0 Z dφ(k)e−Rd4x Lef f = Y k≤Λ0 Z
dφ(k)e−Rd4x (L+sum of connected diagrams).
(2.7)
The extra terms added to the Lagrangian compensate for the integration of modes with Λ0 ≤ k ≤ Λ. There are an infinite number of effective Lagrangians, for all the
different Λ0’s and they all form the renormalisation group flow [15].
2.4
QCD and global symmetries
In section 2.1 the QCD Lagrangian was introduced as well as the SU (3)cgauge
symmetry. The global symmetries of the Lagrangian play an important role as well, especially at low energies, where the dynamics are determined by the symmetries of the QCD vacuum.
A sensible starting point for studying the low energy regime of QCD and its
symmetries is to only consider the three lighter flavours of quarks, (u, d, s) and also to assume that those quarks are massless (chiral limit). This approximation is valid
because (u, d, s) are considerably lighter than c, t, b quarks and furthermore their masses are much smaller than 1GeV, the scale at which the lightest hadrons (which are not Goldstone bosons) are formed. For convenience (2.1) can be rewritten in terms of its chiral components3, so that the symmetries that we wish to study
become manifest. The chiral Lagrangian in the chiral limit is given by
LQCD =
X
l=u,d,c
qL,l(iγµDµ)qL,l+ qR,l(iγµDµ)qR,l−14GaµνGaµν. (2.8)
By simple inspection, the Lagrangian4 (2.8) has a global SU (3)L× SU (3)R flavour
symmetry since the left and right handed components can be independently SU (3) “rotated”. The SU (3)L transformation is the following
qL→ qL0 = exp " −i 8 X α=1 θLαλα 2 # qL,
where θLα are parameters of the SU (3) group. The same applies for the right handed fields, for different parameters θR
α. By investigating further the symmetries
of the lagrangian, it can be verified that the right and left handed fields can be rotated by the same phase
qL→ qL0 = e−iφqL, qR→ qR0 = e−iφqR, (2.9)
without altering the lagrangian. This is the baryon U (1)B symmetry. Also, the
lagrangian remains unchanged when the left handed fields are rotated by a phase and the right handed are rotated by minus the same phase
qL→ qL0 = e−iaqL, qR→ q0R= eiaqR. (2.10) This symmetry is the axial U (1)Asymmetry of the Lagrangian
To summarise, the QCD Lagrangian, at a classical level, is characterised by a global
SU (3)L× SU (3)R× U (1)B× U (1)Asymmetry. In the field theory level, only the
3The Dirac fields can be rewritten in terms of their chiral components. One way to do that is to use the right and left hand projection operators PL= 1 − γ5, PR= 1 + γ5, which when applied on the Dirac fields return back their chiral components qL,l= PLql.
SU (3)L× SU (3)R× U (1)B is a symmetry of the theory. The U (1)A singlet vector
axial current is not conserved and the symmetry is called anomalous5.
In nature the quarks are not really massless and therefore mass terms should be added to (2.1)
LQCDm= X
l=u,d,s
¡
mlqL,lqR,l+ mlqR,lqL,l¢. (2.11) These mass terms mix the right with the left handed fields and therefore explicitly break SU (3)L× SU (3)R symmetry. The U (1)B symmetry is not affected by the
presence of mass terms and therefore is always conserved. If it is assumed that the masses ml are not only finite but equal as well, then the symmetry of the
Lagrangian is enhanced to SU (3)V × U (1)B6. In the case where the masses are
equal and small the chiral symmetry can be considered an approximate symmetry of the Lagrangian.
Of course, the assumption that the three lightest quarks have equal masses is not very good. A better approximation is to assume that the up and down quarks have equal masses and the strange quark has an infinite mass, which is more realistic. Then the chiral symmetry would be SU (2)L× SU (2)R.